Abstract: (This is a sequel to my talk on Tuesday, but I will strive to make it accessible to anyone who couldn’t make it on Tuesday.) On Tuesday I presented a construction that I will begin this talk by summarizing briefly. Then we will examine some applications, most of which are large-cardinal lower bounds for the Ramsey Property of definable sets relative to special classes of coideals. I don’t intend to say much about the upper bounds, but these results complete several equiconsistencies.

If time allows, I will advertise a theorem of Hrušák–Meza-Alcántara–Thümmel–Uzcátegui about Borel ideals on the integers and a related open question. If time is very generous, I will sketch their proof, which is completely combinatorial.